Thanks, John. As Sergeraert says, he hasn't proved his triangulations actually are homeomorphic to $CP^n$! It's still a surprise that explicit triangulations are apparently not known, but on first sight, Tyler's idea looks sound to me. It shows the problem is harder than it looks.

If there are no repeated factors of the polynomial modulo $p$ then certainly $p$ does not divide the discriminant.

Recommendation seconded!

Thanks Ian, that's very interesting; it's a shame I can only tick one response.

You are taking limits as $N\to\infty$. In the sum $\sum_{w\ne0}a_{s+Nw}$ the terms thin out as $N$ increases. Assuming $a_n\to0$ quickly enough as $n\to\pm\infty$ then $\sum_{w\ne0}a_{s+Nw}$ will tend to $0$.

Thanks Andy, that's exactly the sort of thing I was looking for.